Harmonic interpolating sequences, L p and BMO
Annales de l'Institut Fourier, Tome 28 (1978) no. 4, p. 215-228
Soit (z ν ) une suite de points du demi-plan supérieur ; si, pour p tel que 1<p, et pour toute suite (a ν ) νN dans p (N) il existe une fonction f, intégrale de poisson d’une fonction de L p (R) qui vérifie :yν1/pf(zν)=aν,ν=1,2,...(*)alors nous montrons que (z ν ,νN) est une suite d’interpolation pour H . De même, si on fait l’hypothèse qu’il existe une solution f, intégrale de Poisson d’une fonction de BMO qui vérifie (*) avec p=+ et (a ν ) dans (N), (z ν ) est encore une suite d’interpolation pour H .Un théorème un peu plus général est prouvé et on donne un contre-exemple dans le cas où p1.
Let (z ν ) be a sequence in the upper half plane. If 1<p and ifyν1/pf(zν)=aν,ν=1,2,...(*)has solution f(z) in the class of Poisson integrals of L p functions for any sequence (a ν ) p , then we show that (z ν ) is an interpolating sequence for H . If f(z ν )=a ν , ν=1,2,... has solution in the class of Poisson integrals of BMO functions whenever (a ν ) , then (z ν ) is again an interpolating sequence for H . A somewhat more general theorem is also proved and a counterexample for the case p1 is described.
@article{AIF_1978__28_4_215_0,
     author = {Garnett, John B.},
     title = {Harmonic interpolating sequences, $L^p$ and BMO},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {28},
     number = {4},
     year = {1978},
     pages = {215-228},
     doi = {10.5802/aif.721},
     zbl = {0377.46044},
     mrnumber = {80g:30024},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1978__28_4_215_0}
}
Garnett, John B. Harmonic interpolating sequences, $L^p$ and BMO. Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 215-228. doi : 10.5802/aif.721. http://www.numdam.org/item/AIF_1978__28_4_215_0/

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